Thermally-driven atmospheric escape: Transition from hydrodynamic to Jeans escape Alexey N. Volkov1, Robert E. Johnson1,2, Orenthal J. Tucker1, Justin T. Erwin1 1Materials Science and Engineering, University of Virginia, Charlottesville, VA 22904-4745 2Physics Department, NYU, NY, NY 10003-6621 Abstract Thermally-driven atmospheric escape evolves from an organized outflow (hydrodynamic escape) to escape on a molecule by molecules basis (Jeans escape) with increasing Jeans parameter, the ratio of the gravitational to thermal energy of molecules in a planet’s atmosphere. This transition is described here using the direct simulation Monte Carlo method for a single component spherically symmetric atmosphere. When the heating is predominantly below the lower boundary of the simulation region, R0, and well below the exobase, this transition is shown to occur over a surprisingly narrow range of Jeans parameters evaluated at R0: λ0 ~ 2-3. The Jeans parameter λ0 ~ 2.1 roughly corresponds to the upper limit for isentropic, supersonic outflow and for λ0 >3 escape occurs on a molecule by molecule basis. For λ0 > ~6, it is shown that the escape rate does not deviate significantly from the familiar Jeans rate evaluated at the nominal exobase, contrary to what has been suggested. Scaling by the Jeans parameter and the Knudsen number, escape calculations for Pluto and an early Earth’s atmosphere are evaluated, and the results presented here can be applied to thermally-induced escape from a number of solar and extrasolar planetary bodies. 1 Introduction Our understanding of atmospheric evolution is being enormously enhanced by extensive spacecraft and telescopic data on the outer solar system bodies and on exoplanets. The large amount of data on Titan’s atmosphere from the Cassini spacecraft led to the use of models for atmospheric escape that predicted rates that differed enormously (Johnson 2009; Johnson et al. 2009). This disagreement was due to a lack of a kinetic model for how atmospheric escape changes in character from evaporation on a molecule by molecule basis to an organized flow, referred to as hydrodynamic escape, a process of considerable interest early in the early stages of the evolution of a planet’s atmosphere (e.g., Watson et al. 1981; Hunten 1982; Tian et al. 2008) and to the evolution of exoplanet atmospheres (e.g., Murray-Clay et al. 2009). The transition from hydrodynamic to Jeans escape is described in terms of the local Jeans parameter, λ = U(r)/kT(r), where U(r) is a molecule’s gravitational energy at distance r from the planet’s center, T is the local temperature, and k is the Boltzmann constant (e.g., Chamberlain and Hunten 1987; Johnson et al. 2008). In Hunten (1982) it was suggested that if λ decreases to ~2 above the exobase, then the escape rate is not too different from the Jeans rate, but if it becomes ~2 near or below the exobase, hydrodynamic escape can occur. Since it was assumed that the transition region from Jeans to hydrodynamic escape occurred over a broad range of λ, an intermediate model, referred to as the slow hydrodynamic escape (SHE) model, was developed to describe outflow from atmospheres such as Pluto’s with exobase values λ~10 (Hunten and Watson 1982; McNutt 1989; Krasnopolsky 1999; Tian and Toon 2005; Strobel 2008a). In this model, based on that of Parker (1964) for the solar wind, the continuum fluid equations accounting for heat conduction are solved to an altitude where the flow velocity is still smaller than the speed of sound, and then asymptotic conditions on the temperature and density 2 are applied. Unfortunately, this altitude is often above the nominal exobase. This model, subsequently applied to even Titan (λ~40), could overestimate the mass loss rate (Tucker and Johnson 2009; Johnson 2010). Other recent models simply couple the Jeans rate to continuum models at the exobase, even when the escape rates are quite large (Chassefiere1996; Tian et al. 2009), or they couple to a modified Jeans rate (Yelle 2004; Tian et al. 2008). Here the transition from Jeans to hydrodynamic escape from a planetary body is described via a kinetic model. Escape from an atmosphere can be calculated from the Boltzmann equation, which can describe both continuum flow and rarefied flow (e.g., Chapman and Cowling 1970) at large distances from a planet. Here we use the Direct Simulation Monte Carlo (DSMC) method (e.g. Bird 1994) to simulate thermally-driven flow in a one-dimensional radial atmosphere. For escape driven by heat deposited below the lower boundary of the simulation region, R0, we show that the transition occurs over a surprisingly narrow range of Jeans parameters evaluated at R0 (λ0 ~2 – 3). That is, hypersonic outflow drives escape for λ0 ≤ ~ 2, but above λ0 ~ 3 hypersonic flow does not occur, even at large distances from the exobase, and for λ0 > ~6 the escape rate is close to the Jeans escape rate. Following the description of the DSMC model, results for a range of λ0 are presented with emphasis on the transition region. 2. DSMC simulations A kinetic model for calculations of the structure of the upper atmosphere and the escape rate is based on the Boltzmann kinetic equation. A DSMC method (Bird 1994), which is a stochastic method for numerical solution of problems based on the Boltzmann equation, is used here to simulate a spherically-symmetric, single component atmosphere supplied by out gassing from a surface at radius R0 . This can be the actual surface with a vapor pressure determined by the solar insolation or a virtual surface in the atmosphere, above which little additional heat is 3 deposited and at which the density and temperature are known. In such a model, escape is driven by thermal conduction and heat flow from below R0 . In DSMC simulations, the real gas is simulated by means of large number of representative molecules of mass m. Trajectories of these molecules are calculated in a gravity field and subject to mutual collisions. It is readily shown that for a velocity independent cross section, the Boltzmann equations, and, hence, the results presented here, can be scaled by two parameters: the source values of the Jeans parameter, λ0, 1/2 -1 and a Knudsen number, Kn0 = l0 /R0 where l0 =(2 n0σ) is the mean free path of molecules at the lower boundary with σ the collision cross section and n0 the number density on lower boundary. The Knudsen number often discussed for a planet’s atmosphere is Kn(r) = l / H, where l and H are the local mean free path of molecules and the atmospheric scale height at radial distance r. Since Kn(R0) = λ0 Kn0, it could be used, instead of Kn0, as one of the two scaling parameters. Simulations were conducted on a non-homogeneous mesh and the number of simulated molecules in a cell was varied over a wide range (e.g., Volkov et al. 2010). Results are presented for the hard sphere collisions, but comparisons made using variable hard spheres and forward directed collision models (e.g.,Tucker and Johnson 2009) gave similar results. In the kinetic model, the flow at each r is described in terms of a velocity distribution function, f (r,v||,v⊥ ,t) , where v|| and v⊥ are velocity components parallel and perpendicular to radial direction and t is time. Macroscopic parameters are calculated for molecular quantities Ψ(r,v|| ,v⊥ ) using the +∞∞ integral operator Ψ = 2π Ψ(r,v ,v ) f (r,v ,v ,t)v dv dv , e.g., number density, n ∫∫ || ⊥ || ⊥ ⊥ ⊥ || −∞ 0 2 ( Ψ = 1), radial flow velocity, u (Ψ = v|| / n), parallel, T|| (Ψ = m(v|| − u) /(nk) , and 4 2 perpendicular, T⊥ ()Ψ = mv⊥ /(2nk ), temperatures with T = (T|| +2T⊥ )/3. The molecules have gravitational energyU (r) = −GMm / r , where M is the planet’s mass and G is the gravitational constant. The mass of the gas above R0 is assumed to be much smaller than M so that self- gravity is neglected. The velocity distribution in the boundary cell at r=R0 is maintained to be Maxwellian at a fixed n0 and T0 , and zero gas velocity. The exit boundary at r = R1 is placed far enough from R0 so the gas flow above this boundary can be approximately treated as collisionless. A molecule crossing R1 with velocities v|| and v⊥ will escape if 1/ 2 2 2 v > (−2U (R1) / m) , where v = v|| + v⊥ , while a molecule with a smaller v will return to R1 with − v|| and v⊥ . Since collisions can modify the flow even at relatively large r, the effect of the position of R1 was studied (Volkov et al. 2010) and R1 is chosen to be sufficiently large in order to eliminate the effect of upper boundary in the flow region described. The simulations were carried out using the following fixed parameters, m, σ, T0 and R0 with λ0 and Kn0 varied by changing M and n0. However, the results scale with λ0 and Kn0 in a sense that any two flows with different m, σ, T0, R0, M, and n0 represent the same flow in a dimensionless form, if the λ0 and Kn0 are the same. Simulations are initiated by ‘evaporation’ from the cell at R0 until steady flow is reached at large t. Steady-state flow is assumed to have occurred when the number flux 4πr 2n(r)u(r) = Φ varies by less that ~1% across the domain, where Φ is the escape rate. The Jeans escape rate, ΦJeans, is defined by the upward flux of molecules with speeds 1/ 2 v ≥ (−2U (r) / m) at the nominal exobase rexo, typically determined by the scale height [l(rexo) = H(rexo)] at large λ0 or by the curvature [l(rexo) = rexo] at small λ0.